In this note we generalize the well-known Wigner’s unitary-antiunitary theorem. For smooth normed spaces X and Y and a surjective mapping \(f :X\rightarrow Y\) such that \(|[f(x),f(y)]|=|[x,y]|\), \(x,y\in X\), where \([\cdot ,\cdot ]\) is the unique semi-inner product, we show that f is phase equivalent to either a linear or an anti-linear surjective isometry. When X and Y are smooth real normed spaces and Y is strictly convex, we show that Wigner’s theorem is equivalent to \(\{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}\), \(x,y\in X\).

在本文中，我们归纳了著名的维格纳的W反一unit定理。对于光滑的范数空间X和Y以及射影映射\（f：X \ rightarrow Y \）使得\（| [f（x），f（y）] | = | [x，y] | \），\ （x，y \ in X \），其中\（[\ cdot，\ cdot] \）是唯一的半内积，我们证明f相等于线性或反线性等距等距线。当X和Y是光滑实范空间且Y严格凸时，我们证明Wigner定理等于\（\ {\ Vert f（x）+ f（y）\ Vert，\ Vert f（x）-f（ y）\ Vert \} = \ {\ Vert x + y \ Vert，\ Vert xy \ Vert \} \），\（x，y \ in X \）。